Solving Two-Step Equations | Algebra Equations

Summary

In the 'Math with Mr. J' video, the focus is on solving two-step equations. Mr. J demonstrates how to isolate variables by reversing operations, ensuring equation balance. Examples include undoing subtraction and division to isolate 'x' in '2x - 6 = 10', leading to x = 8, and 'R/5 + 8 = 11', resulting in R = 15. The video also covers handling parentheses and variables on different sides of the equation, emphasizing the importance of checking solutions in the original equations.

Takeaways

• 🔢 The main goal in solving equations is to isolate the variable.
• ⚖️ To maintain balance, whatever operation is done to one side of the equation must be done to the other side.
• 🔄 The process involves reversing the order of operations to isolate the variable.
• ➕ To eliminate subtraction on one side, add the opposite on both sides.
• ➗ To remove multiplication, use division as the opposite operation.
• 🔄 For equations like '2x - 6 = 10', first add 6 to both sides to eliminate the subtraction, then divide by 2 to solve for x.
• 🔄 In equations with division and addition, such as 'r/5 + 8 = 11', subtract 8 from both sides first, then multiply by 5 to solve for r.
• 🔄 When the variable is on the right side, like in '7 = 16 - 3e', subtract the constant from both sides to move the variable to the left.
• 🔄 Parentheses can be handled by dividing both sides by the coefficient outside the parentheses to simplify the equation.
• 🔄 Always check the solution by plugging the isolated variable back into the original equation to ensure accuracy.

Q & A

What is the main goal when solving equations with variables?

The main goal is to isolate the variable, getting it by itself to solve the equation.

Why is it important to perform the same operation on both sides of an equation?

It is important to perform the same operation on both sides to keep the equation balanced.

In the first example, what is the first step to isolate the variable 'x'?

The first step is to add 6 to both sides of the equation to eliminate the subtraction of 6 on the left side.

How does adding 6 to both sides of the equation 2x - 6 = 10 help in solving for x?

Adding 6 to both sides results in 2x = 16, which simplifies the equation and brings us closer to isolating x.

What is the reverse operation of multiplication used in the script?

The reverse operation of multiplication is division, which is used to isolate the variable by making the coefficient equal to 1.

For the equation R / 5 + 8 = 11, how do you reverse the operation of addition?

To reverse the addition, you subtract 8 from both sides of the equation to isolate the term with the variable R.

What is the purpose of multiplying both sides of an equation by 5 when solving for R in the equation R / 5 + 8 = 11?

Multiplying both sides by 5 reverses the division by 5, which helps to isolate the variable R.

In the equation 7 = 16 - 3e, how do you handle the negative sign in front of the variable?

You divide both sides by -3 to isolate the variable e, reversing the multiplication by -3.

Why is it necessary to divide both sides by 2 in the equation 2(y - 8) = 24?

Dividing both sides by 2 undoes the multiplication by 2 outside the parentheses, simplifying the equation to y - 8 = 12.

How does adding 8 to both sides of the equation y - 8 = 12 help in isolating y?

Adding 8 to both sides cancels out the -8 on the left side, leaving y by itself on the left side of the equation.

What is the final step to verify the solution to an equation?

The final step is to plug the solution back into the original equation to see if it satisfies the equation and yields the correct result.

Outlines

00:00

📘 Solving Two-Step Equations

This segment of the video introduces the concept of solving two-step equations with Mr. J. The process involves isolating the variable to solve the equation, ensuring that any operation performed on one side of the equation is mirrored on the other to maintain balance. The video demonstrates this with the equation 2x - 6 = 10, where Mr. J reverses the operations by first adding 6 to both sides to eliminate the subtraction, and then dividing by 2 to isolate x, resulting in x = 8. The solution is verified by substituting the value back into the original equation.

05:03

📗 Advanced Two-Step Equation Techniques

In the second part, the video script delves into more complex two-step equations, such as R/5 + 8 = 11 and 7 = 16 - 3e. The method involves reversing the operations to isolate the variable. For R/5 + 8 = 11, Mr. J subtracts 8 from both sides and then multiplies by 5 to solve for R, finding R = 15. For 7 = 16 - 3e, the script shows subtracting 16 from both sides and then dividing by -3 to solve for e, resulting in e = 3. Each solution is checked by substituting the found value back into the original equation to confirm its correctness.

Keywords

💡Two-step equations

Two-step equations are algebraic equations that require two operations to solve for the variable. In the video, these equations are used to demonstrate the process of isolating the variable by reversing the operations. For example, the equation '2x - 6 = 10' is a two-step equation that involves first adding 6 to both sides to eliminate the subtraction, and then dividing by 2 to solve for x.

💡Isolate the variable

Isolating the variable means getting the variable alone on one side of the equation, which is a crucial step in solving algebraic equations. The video emphasizes this by showing how to reverse operations to achieve this, such as adding or subtracting terms to cancel out numbers on the same side of the equation.

💡Reverse order of operations

The reverse order of operations is a strategy used in the video to solve equations by undoing the operations applied to the variable. For instance, if an equation has a term '2x', the reverse operation would be dividing by 2 to isolate 'x'. This concept is fundamental to the video's teaching of solving two-step equations.

💡Balanced equation

A balanced equation is one where the operations performed on one side are mirrored on the other side to maintain equality. The video script mentions that whatever operation is done to one side must be done to the other to keep the equation balanced, which is essential for accurate problem-solving.

💡Subtraction

Subtraction is one of the basic arithmetic operations used in the video to manipulate equations. For example, in the equation '2x - 6 = 10', adding 6 to both sides is used to eliminate the subtraction on the left side, helping to isolate the variable.

💡Division

Division is another arithmetic operation used in the video to solve equations. After performing subtraction to isolate the term with the variable, division is used to eliminate the coefficient of the variable, as shown when dividing both sides by 2 in the equation '2x = 16' to solve for x.

💡Multiplication

Multiplication is used in the video to demonstrate the opposite operation of division when solving equations. For instance, if a term is divided by a number, multiplying by that number is the reverse operation to isolate the variable, as seen when multiplying both sides by 5 in the equation 'R/5 + 8 = 11'.

💡Parentheses

Parentheses are used in algebraic expressions to group terms and indicate that the operations inside them should be performed first. In the video, parentheses are part of the equation '2(y - 8) = 24', and the video explains how to deal with them by dividing by the coefficient outside the parentheses to simplify the equation.

💡Variable

A variable is a symbol, often a letter, that represents an unknown value in an equation. The video's main focus is on solving for variables in two-step equations, using various algebraic techniques to isolate them and determine their values.

💡Coefficient

A coefficient is a numerical factor that multiplies a variable in an algebraic expression. In the video, coefficients are part of the equations that need to be reversed or eliminated to isolate the variable, such as the '2' in '2x' which is reversed by dividing by 2.

Highlights

Introduction to solving two-step equations.

Goal of isolating the variable in an equation.

Principle of maintaining equation balance by performing the same operation on both sides.

Solving the first equation: 2x - 6 = 10 by reversing operations.

Adding 6 to both sides to eliminate the -6 on the left side.

Dividing both sides by 2 to isolate x.

Verification of the solution x = 8 by plugging it back into the original equation.

Solving the second equation: R/5 + 8 = 11 using reverse order of operations.

Subtracting 8 from both sides to isolate R/5.

Multiplying both sides by 5 to solve for R.

Verification of the solution R = 15 by checking the original equation.

Approach to solving equations with variables on the right side, such as 7 = 16 - 3e.

Eliminating the constant 16 by subtracting 16 from both sides.

Dividing both sides by -3 to isolate e.

Verification of the solution e = 3 by substituting it into the original equation.

Handling equations with parentheses, like 2(y - 8) = 24.

Dividing both sides by 2 to eliminate the multiplication outside the parentheses.

Adding 8 to both sides to isolate y.

Verification of the solution y = 20 by plugging it back into the original equation.

Summary of the method for solving two-step equations.

Transcripts

welcome to math with mr. J in this video

I'm going to cover how to solve two-step

equations we have for example problems

on your screen there that we're going to

go through together in order to get this

down now remember when we have an

equation with a variable our goal is to

isolate that variable or get it by

itself in order to solve and we also

need to remember whatever we do to one

side we must do to the other side of the

equation we have to keep it balanced so

let's jump right into the number one and

solve some two-step equations so for

number one we have 2x minus 6 equals 10

so again we want to isolate that X get

it by itself so I like to think of it as

we need to reverse the order or undo

this side of the equation so we get that

X by itself and we're going to use the

reverse order of operations in order to

do so so we have 2 times that X and then

we subtract a 6 so reverse order of

operations this subtraction of 6 needs

to come first so how do we get rid of

that 6 from the left side well we can

add 6 that will cancel those sixes out

or give us a 0 so remember whatever we

do to one side we have to do to the

other so if we add 6 to the left we need

to add 6 to the right a negative 6 or

minus 6 plus 6 gives us that 0 and 10

plus 6 is 16 so on the left side we're

left with 2 times X or 2x so we don't

have the variable completely isolated

yet but we're almost there so we have 2

times X so how do we get rid of that 2

we need to either make it a 0 or a 1 so

the opposite of multiplying by 2 would

be dividing by two that would give us

one X on that side which is the same as

just X so let's divide both sides by two

and that leaves us with x equals 16

divided by 2 is 8 now let's plug in that

8 into the original original equation

and see if we get the correct answer so

2 times 8 minus 6 equals 10 it's always

a good idea to see if that answer works

out 2 times 8 is 16 minus 6 does give us

that 10 so we have the correct answer x

equals 8 so for number two we have R

divided by 5 plus 8 equals 11 so we need

to get that R by itself so let's do the

reverse order of operations to undo the

left side of the equation so let's get

rid of that 8 first so we have plus 8 so

the opposite let's subtract 8 from both

sides to begin to isolate the R so a

positive 8 and a negative 8 there minus

8 gives us zero and 11 minus 8 gives us

3 so on the left side we're left with R

divided by 5 so let's get rid of the 5

from the left side what's the opposite

of divided by 5 dividing by 5 well

multiplying by 5 so let's multiply both

sides by 5 by 5 by 5 and we get R equals

well 3 times 5 is 15 we isolated the

variable and it equals 15 so on the left

hand side I just want to mention we had

R divided by 5 that last step and we

times by 5 which would technically give

us R over 1 or

are divided by one which is just our

this is isolating the variable right

here if you get to multiplying that

variable by one or dividing that

variable by one so let's plug in that 15

and see if we get the correct answer

here so I'm running out of room a little

bit I'll fit it in here so 15 divided by

5 is 3 bring down an hour 8 and we end

up with 3 plus 8 which gives us the 11

we want it so let's go over to number 3

here where we have 7 equals 16 minus 3e

so the equation looks a little different

than numbers 1 & 2 we have the variable

on the right-hand side but it's the same

exact thing that we did for numbers 1 &

2 so we need to isolate that e so undo

that right side of the problem so let's

get rid of the 16 first so we have a

positive 16 on the right hand side so

the opposite would be subtracting 16 in

order to get rid of it let's do minus 16

on the left hand side as well so 16

minus 16 gives us at 0 7 minus 16 gives

us a negative 9 we're left with negative

3 e on the right side so that's

multiplication so we need to do the

opposite of multiplication in order to

get the e by itself so let's divide both

sides by negative 3 negative 9 divided

by negative 3 gives us a positive 3 and

we're left with E over 1 which is the

same thing as just E we isolated the

variable so e equals 3 let's plug it

back in and see if that works

three times three is nine

bring down our 16 16 minus nine does

give us that seven that we are looking

for on the left-hand side of that

equation so we were correct e equals

three and lastly number four so we have

some parentheses in this one and we need

to get Y by itself or isolate the

variable Y so we have two times

parenthesis Y minus eight and

parenthesis equals 24 so we need to do

the opposite remember we need to undo

that side the left-hand side of the

equation and we're going to actually

divide both sides by two to undo that

two that is outside of the parenthesis

so two divided by two is one that gives

us one outside of the parenthesis there

which is just going to leave us with y

minus eight because anything times one

is just that number or expression itself

so we just have Y minus eight and then

24 divided by two is 12 so now we have Y

minus 8 equals 12 so we need to get rid

of that minus eight undo that part of

the left hand side of the equation in

order to isolate the Y so we need to add

8 to both sides in order to isolate the

Y so a minus 8 and a plus 8 gives us a 0

those cancel out so we're left with Y

and 12 plus 8 gives us 20 so y equals 20

let's plug it back into the equation to

see if this gives us the answer 24 that

we're looking for

20 minus 8 is 12 bring down the 2

outside of the parentheses which means

multiplication and 2 times 12 does give

us that 24 that we wanted so there you

have it there's how you solve two-step

equations hopefully that helped thanks

so much for watching

until next time peace